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Math Help - ANother model

  1. #1
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    ANother model

    Hi people I dont know if anyone can help, but I'm working on some models and I'm having a bit of difficulty with this one..

    <br />
\dot S(t) \equiv \frac{{dS}}<br />
{{dt}} =  - \alpha SI<br />

    where is alpha is a const.

    rate which no. of those infected changes depending on suseptibles getting ill and other recovering,

    This process is described by

    <br />
\dot I(t) \equiv \frac{{dI}}<br />
{{dt}} = \alpha SI - \beta {\rm I}<br />

    where beta is a const.

    So i need to know how to form the differential equation for I(S) and given that I=1 when S=N, find solution for I(S).
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  2. #2
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    Quote Originally Posted by reivera View Post
    Hi people I dont know if anyone can help, but I'm working on some models and I'm having a bit of difficulty with this one..

    <br />
\dot S(t) \equiv \frac{{dS}}<br />
{{dt}} =  - \alpha SI<br />

    where is alpha is a const.

    rate which no. of those infected changes depending on suseptibles getting ill and other recovering,

    This process is described by

    <br />
\dot I(t) \equiv \frac{{dI}}<br />
{{dt}} = \alpha SI - \beta {\rm I}<br />

    where beta is a const.

    So i need to know how to form the differential equation for I(S) and given that I=1 when S=N, find solution for I(S).
    \frac{dI}{dS} = \frac{dI}{dt} \, \cdot \frac{dt}{dS} = (\alpha S I - \beta I) \, \left( -\frac{1}{\alpha S I} \right) = \frac{\beta - \alpha S}{\alpha S},  \, I \neq 0,


    = \frac{\beta}{\alpha} \, \frac{1}{S} - 1,


    subject to the boundary condition 1 = I(N).


    I get I = \frac{\beta}{\alpha} \ln \left( \frac{S}{N} \right) - S + N + 1 as the solution.

    Question for you: This solution is valid provided I \neq 0. Is this condition always met?
    Last edited by mr fantastic; March 4th 2008 at 03:30 PM.
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  3. #3
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    Your answer seems fine to me, I dont think there is any reason I = 0 is a problem, in this model I(t) is the number currently infected...
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